Flaky paint after removing wallpaper

Hi
I have recently removed wallpaper from my living room, hallway and staircase as I want to paint the walls.
However I am left with a lot of patches where the old paint underneath the wallpaper has come off.
I need advice what would be the easiest and quickest way to paint the walls as I have a lot of walls to paint.

Would I need to skim over the walls or can I paint over it after sanding it down?

Thank you